The resultant of ⃗⃗P and ⃗⃗Q is ⃗⃗R. If ⃗⃗Q is doubled, ⃗⃗R is also doubled; when Q is reversed, ⃗⃗R is again doubled. Find P: Q: R.
Answers
Answer:
Let θ be the angle between
P
and
Q
. Then
R
2
=∣
P
+
P
∣=P
2
+Q
2
+2PQcosθ ...(i)
If
Q
is doubled,
R
is doubled. That means, the magnitude of resultant of 2
Q
and
P
is
(2R)
2
=P
2
+(2Q)
2
+2P(PQ)cosθ
This yields 4
R
2
=P
2
+4Q
2
+4PQcosθ ...(ii)
When
Q
is reversed,
R
is doubled. Hence, the magnitude of resultant of
P
and (−
Q
) is 2R.
Then (2R)
2
=P
2
+Q
2
+2PQcos(180−θ) ...(iii)
This yields 4R
2
=P
2
+Q
2
−2PQcosθ ...(iii)
(i) - (ii) yields PQcosθ=
4
−3R
2
...(iv)
(i) + (iii) yields P
2
+Q
2
=
2
5R
2
...(v)
(ii) + (iv) yields P
2
+4Q
2
=7R
2
...(vi)
Solving (v) and (vi), we obtain Q=
2
3
chal
R and P=R.
Hence, P:Q:R=
2
:
3
:
2